Elementary topoi

As the prototypical category, $\mathbf{Set}$ has many properties which make it special amongst categories. From the point of view of mathematical logic, one such property is that $\mathbf{Set}$ has enough structure to "properly" formalise logic. However, we could ask what it might mean to formalise logic in another category $\mathbf{C}$. The notion of an (elementary) topos distills the essential features of $\mathbf{Set}$ which allow us to do this. This expository report defines topoi, and describes the development of first-order logic and set theory within a topos.